This is not an infinite decimal. The digit 1 is somewhere out there.

But this is not a compelling argument to somebody in this situation. While correct, it feels identical to saying "it just is".

The whole numerical representation scheme really is just a man made system. If you dig deep enough the veneer disappears. This is especially noticeable when you start to see things like numbers that are finite in decimal but have infinite repeating patterns binary.

Though you could treat it as an infinite series in a similar way:

0.9 -> 0.99 -> 0.999 -> ... -> ?

0.1 -> 0.01 -> 0.001 -> ... -> ?

The first is an infinite series:

  9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...

The second is not?!

I mean you could if you want to:

1/10 + -9/100 + -9/1000 + -9/10000 + ...

But I just meant them as series, not necessarily as sums.

Or

  1 - 9/10 - 9/100 - ...

or

  1 - ( 9/10 + 9/100 + ... )

So it becomes circular:

  0.00...1 = 1 - 0.999...

(In math a series is a sum:

https://en.m.wikipedia.org/wiki/Series_(mathematics)

Woops, you're totally right, I think I should've said "sequence".

Because unlike 0.99..., 0.00...1 is not a Real number.

The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

Then the question is, is 0.00....1 equal to zero? We use the definition of equality above and we say yes.

EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.

It is not.

The decimal representation of a Real number has to be indexed by Natural numbers, i.e. every decimal digit[n] has a well-defined index n which is a Natural number. Infinity is not a Natural number, so 0.00...1 is not a Real number either.

This is usually described mathematically by saying that the representation of a decimal has to be countable, whereas the number 0.000...1 is not a countable representation.

I will say though that if your explanation of 0.999... = 1.0 requires that you explain the distinction between countable and uncountable infinities, that's a big ask for most lay people.

Could you please explain a little more what "not a countable representation" means?

A countable set is one where you can reach any element in a finite amount of steps. See:

https://en.wikipedia.org/wiki/Countable_set

If you're saying we never get from the initial 0 to the trailing 1 in a finite number of steps, that's true and that's what DavidVoid is saying. But I haven't made the list of digits uncountably large by adding one element. Instead I put that element in a transfinite position in the ordering and I have to watch out for weird consequences.

I'm no expert, but countability doesn't depend on how the set is ordered. It depends on whether the elements can be placed in 1:1 correspondence with the integers. 1,0,0,... has a countable number of elements, and so does 0,0,0,...,1. They can be put in 1:1 correspondence with each other. This definition of countability is described in your link.

I meant to ask, does "countable representation" some kind of detailed definition that I can look at?

It's not the set of digits you use which is uncountable, it's the representation itself.

I think it's just another way to word DavidVoid's explanation a few comments up.

Ok. (a) I think you're saying that 0.00 ... 1 is not a real number. (b) Do we agree that lim n->inf 10^-n is a real number? (I.e. zero?) (c) Then you're saying that limit in "(b)" is not a reasonable definition of the string "0.00 ... 1". Is that correct?

That is correct.

The limit of lim n->inf 10^-n is exactly 0, it is not 0.00...1.

Since we have no common definition of what 0.00...1 might possibly mean, let's say we agree.

0.0000... is the repeating decimal representation of zero.

What is 0.00...1 times 34?

0.00...34

Is 0.00...1 times 34 equal to 0.00..1 times 3.4?

I think the answer to that question depends on the axioms you are using.